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Vocoder Analysis

The definitions of phase delay and group delay apply quite naturally to the analysis of the vocoder (``voice coder'') [21,26,54,76]. The vocoder provides a bank of bandpass filters which decompose the input signal into narrow spectral ``slices.'' This is the analysis step. For synthesis (often called additive synthesis), a bank of sinusoidal oscillators is provided, having amplitude and frequency control inputs. The oscillator frequencies are tuned to the filter center frequencies, and the amplitude controls are driven by the amplitude envelopes measured in the filter-bank analysis. (Typically, some data reduction or envelope modification has taken place in the amplitude envelope set.) With these oscillators, the band slices are independently regenerated and summed together to resynthesize the signal.

Suppose we excite only channel $ k$ of the vocoder with the input signal

$\displaystyle a(nT) \cos(\omega_knT), \qquad n=0,1,2,\ldots $

where $ \omega_k$ is the center frequency of the channel in radians per second, $ T$ is the sampling interval in seconds, and the bandwidth of $ a(nT)$ is smaller than the channel bandwidth. We may regard this input signal as an amplitude modulated sinusoid. The component $ \cos(\omega_k n T)$ can be called the carrier wave, while $ a(nT)\geq 0$ is the amplitude envelope.

If the phase of each channel filter is linear in frequency within the passband (or at least across the width of the spectrum $ A(e^{j\omega T})$ of $ a(nT)$ ), and if each channel filter has a flat amplitude response in its passband, then the filter output will be, by the analysis of the previous section,

$\displaystyle y_k(n) \;\approx\; a[nT - D(\omega_k)] \cos\{\omega_k[nT - P(\omega_k)]\} \protect$ (8.8)

where $ P(\omega_k)$ is the phase delay of the channel filter at frequency $ \omega_k$ , and $ D(\omega_k)$ is the group delay at that frequency. Thus, in vocoder analysis for additive synthesis, the phase delay of the analysis filter bank gives the time delay experienced by the oscillator carrier waves, while the group delay of the analysis filter bank gives the time delay imposed on the estimated oscillator amplitude-envelope functions.

Note that a nonlinear phase response generally results in $ D(\omega_k)\neq P(\omega_k)$ , and $ D(\omega_k)\neq D(\omega_l)$ for $ k\neq l$ . As a result, the dispersive nature of additive synthesis reconstruction in this case can be seen in Eq.(7.8).

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-09-03 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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